The owner of the Rancho Grande has 3028 yd of fencing with which to enclose a rectangular piece of grazing land situated along the straight portion of a river. If fencing is not required along the river, what are the dimensions of the largest area he can enclose?

shorter side yd
longer side yd

What is this area

To find the dimensions of the largest area that can be enclosed with 3028 yards of fencing, we need to consider the shape of the grazing land.

Let's assume that the width (shorter side) of the rectangular grazing land is y and the length (longer side) is x. Since the fencing is not required along the river, the length of the grazing land will be perpendicular to the river.

The perimeter of the grazing land is given by the equation:
2x + y = 3028

To find the dimensions that maximize the area, we can use the concept of calculus.

1. Rearrange the perimeter equation to solve for y:
y = 3028 - 2x

2. Write an equation for the area of the grazing land in terms of x and y:
A = xy

3. Substitute the value of y from step 1 into the area equation:
A = x(3028 - 2x)

4. Expand the equation and simplify:
A = 3028x - 2x^2

Now, to find the maximum area, we need to find the value of x that maximizes the area. This can be done by finding the derivative of the area equation with respect to x, setting it equal to zero, and solving for x.

5. Differentiate the area equation with respect to x:
dA/dx = 3028 - 4x

6. Set the derivative equal to zero and solve for x:
3028 - 4x = 0
4x = 3028
x = 757

7. Substitute the value of x back into the perimeter equation to find the corresponding value of y:
2(757) + y = 3028
1514 + y = 3028
y = 1514

Therefore, the dimensions of the largest area that can be enclosed with 3028 yards of fencing are:
Shorter side (width) = 1514 yards
Longer side (length) = 757 yards

To find the area, we can substitute these dimensions back into the area equation:
Area = 1514 * 757 = 1,146,998 square yards